A Modified Multiple-Mixing-Cell Algorithm for Minimum Miscibility

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Thermodynamics, Transport, and Fluid Mechanics

A Modified Multiple-Mixing-Cell Algorithm for Minimum Miscibility Pressure Prediction with the Consideration of Asphaltene-Precipitation Effect Ruixue Li, and Huazhou Andy Li Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.9b02928 • Publication Date (Web): 31 Jul 2019 Downloaded from pubs.acs.org on July 31, 2019

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Industrial & Engineering Chemistry Research

“

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A Modified Multiple-Mixing-Cell Algorithm for Minimum Miscibility Pressure Prediction with the Consideration of Asphaltene-Precipitation Effect

6 7 8 9 10 11 12 13

Ruixue Li1, 2 and Huazhou Li2*

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15 16 17

2

College of Energy, Chengdu University of Technology, Chengdu, 610059, PR China

School of Mining and Petroleum Engineering, Faculty of Engineering, University of Alberta, Edmonton, Canada T6G 1H9

18 19 20 21 22 23 24

*Corresponding Author: Dr. Huazhou Li

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Associate Professor, Petroleum Engineering

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University of Alberta

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Phone: 1-780-492-1738

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Email: [email protected]

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1 ACS Paragon Plus Environment

Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Abstract

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Minimum miscibility pressure (MMP) is one of the most important design parameters in CO2

32

flooding. Recent experimental studies show that asphaltene precipitation in light oils increases the

33

MMP between injection gas and crude oil. To model the asphaltene-precipitation effect on MMP

34

calculations, we develop a three-phase multiple-mixing-cell (MMC) algorithm by considering

35

three-phase vapor-liquid-asphaltene (VLS) equilibria. This algorithm is developed by modifying

36

the MMC method proposed by Ahmadi and Johns (2011). The three-phase VLS equilibrium

37

calculation algorithm developed by Li and Li (2019) is integrated into the two-phase MMC

38

algorithm. Several example calculations are carried out to demonstrate the performance of our

39

algorithm. The MMPs predicted by our algorithm and those predicted by the two-phase MMC

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algorithm are both compared with the MMPs measured by experiments. Comparison results show

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that MMPs predicted by our algorithm have a better agreement with the experimental results than

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those predicted by the two-phase MMC algorithm.


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1. Introduction

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CO2 flooding becomes a widely applied enhanced oil recovery process in light oil reservoirs [1,

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2]. In the design of CO2 flooding, it is essential to precisely predict the minimum miscibility

47

pressure (MMP) between CO2 and reservoir fluid. At MMP, we can theoretically achieve 100%

48

oil recovery from the reservoir. Asphaltene precipitation is common during CO2 flooding due to

49

the dynamic variations in the composition of reservoir fluids [3, 4]. Recent experimental studies

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reveal that the asphaltene deposition can increase the measured MMPs between the reservoir fluid

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and the injection gas [5]. Therefore, in order to obtain an accurate MMP between CO2 and reservoir

52

fluid, an MMP prediction method considering asphaltene-precipitation effect is required.

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There are many experimental or computational methods that have been developed to determine

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MMPs. The most common experimental method for determining MMP is slim tube test [6]. It is

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considered to be reliable for MMP determination, since it can capture the dynamic flow

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interactions in porous media by considering the phase behavior of reservoir fluids. However, it is

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an expensive and time-consuming test. Moreover, Johns et al. [7] pointed out that slim tube

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experiments may fail to give an accurate MMP due to the limited number of experimental data

59

points and the existence of dispersion phenomenon. Another common experimental method for

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determining MMP is multi-contact experiment. For miscibility achieved via a vaporizing or

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condensing drive during gas flooding, the multi-contact experiment can provide a precise MMP.

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However, the miscibility of most gas flooding processes is achieved via a combined condensing

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and vaporizing drive, making the MMP measured by the multi-contact experiment less accurate

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[8, 9].

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Since the experimental method for MMP determination is expensive and time-consuming, many

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attempts are made to develop computational methods for MMP determination based on equation 3 ACS Paragon Plus Environment


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of state (EOS). There are three main computational methods for MMP determination: slim tube

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compositional simulation, method of characteristics (MOC), and multiple mixing cell (MMC)

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method. Slim tube compositional simulation models the slim tube experiment in a numerical

70

manner. The MMP can be developed at the inflection point on the recovery factor curve as a

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function of pressure [10]. Compared with other computational methods, one of the drawbacks of

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this computational method is its high computational demand. The MOC method is developed to

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analytically solve the c-1 key tie lines (where c represents the number of the component in the

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given mixture) [9, 11-13]. MMP is the minimum pressure where any of the key tie lines has a

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length of zero. However, the MOC method is complex and could easily converge to a wrong key

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tie line [14].

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There are several published MMC approaches. Two MMC approaches are widely used. The first

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approach is a simplification of the slim tube compositional simulation by ignoring the flow

79

equations [15, 16]. In this approach, a series of cells initially filled with oil are connected with

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each other. Then, a specified volume of gas is injected into the first cell. A two-phase vapor-liquid

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equilibrium calculation is required to be conducted on the mixture in the first cell at the given

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temperature/pressure condition. The excess volume will be moved to the second cell and mixed

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with the initial oil in the second cell. The same transfer is made on the following cells. After

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finishing the first injection, another injection will be conducted until a given volume of gas is

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injected. The aforementioned calculations will be conducted at different pressures. The oil

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recovery factor at each pressure should be recorded. Jaubert et al. [15] regarded the pressure, where

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97% oil recovery is achieved, to be MMP. Asphaltene precipitation can be commonly seen during

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gas injection processes [3, 4]. The presence of asphaltene phase will affect the two-phase vapor-

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liquid equilibrium, leading to a change of the MMPs between the injection gas and crude oil. To 4 ACS Paragon Plus Environment

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model asphaltene-precipitation effect on MMPs, Kariman Moghaddam and Saeedi Dehaghani [17]

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developed an MMP determination algorithm by extending the MMC method proposed by Jaubert

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et al. [15]. The main modification is that, after the two-phase vapor-liquid equilibrium calculation,

93

they suggested conducting a two-phase liquid-asphaltene equilibrium calculation on the resulting

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liquid phase. When the asphaltene phase precipitates, it will remain in the cell. In their algorithm,

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the two-phase vapor-liquid equilibrium is used to estimate the amounts of vapor phase and liquid

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phase, while the two-phase liquid-asphaltene equilibrium calculation is performed to yield the

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amounts of asphaltene phase and unprecipitated liquid phase. However, asphaltene precipitation

98

will affect the overall two-phase vapor-liquid equilibrium. A three-phase VLS equilibrium

99

calculation is hence required to accurately capture the three-phase VLS equilibria, instead of

100

conducting separately the two-phase vapor-liquid and the two-phase liquid-asphaltene equilibrium

101

calculations.

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Another MMC approach is proposed by Ahmadi and Johns [18]. In this method, the first contact

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is conducted between two cells that are filled with injection gas and oil. By applying the negative

104

two-phase flash calculation [19, 20], additional two cells will be obtained. The equilibrium gas

105

phase is always put ahead of the equilibrium liquid phase considering that the gas phase moves

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faster than the liquid phase. A number of contacts are conducted by mixing the neighbouring cells

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until all c-1 key tie lines are obtained. The aforementioned calculations should be conducted at

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different pressures. The minimum tie line length is required to be recorded at each pressure. The

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MMP is the pressure where the minimum tie line length becomes zero. This MMC method

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provides a simple way to predict MMP, since it is independent of relative permeability, volume of

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cells, and volume of injected gas. The performance of this method only relies on the robustness of

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two-phase flash calculation algorithm. But currently the original MMC algorithm developed by 5 ACS Paragon Plus Environment


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Ahmadi and Johns [18] cannot accommodate the asphaltene precipitation phenomenon, and

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thereby cannot model the effect of asphaltene precipitation on the MMP.

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In this study, we develop an MMC algorithm considering asphaltene-precipitation effect by

116

modifying the MMC method proposed by Ahmadi and Johns [18]. In the revised MMC algorithm,

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before each contact, we first perform a three-phase VLS equilibrium calculation to check if the

118

asphaltene phase will appear. The three-phase VLS equilibrium calculation algorithm used in our

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algorithm is the one recently developed by Li and Li [21] with the application of the asphaltene-

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precipitation model proposed by Nghiem et al. [22]. If asphaltene phase appears, the asphaltene

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phase is considered to be deposited in the cell containing the liquid phase. Then, the composition

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of the mixture of the fluids in two neighbouring cells will be updated by excluding the asphaltene

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phase. Afterwards, a two-phase flash calculation is conducted on the mixture with the updated

124

composition. Example calculations are conducted for several real fluid mixtures to test the

125

performance of the newly developed algorithm. The MMPs predicted by our algorithm, together

126

with those predicted by the MMC algorithm without considering asphaltene-precipitation effect,

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are both compared with the MMPs measured by slim tube experiments.

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2 Thermodynamic Model

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For two-phase flash calculations, Peng-Robinson (PR) EOS [23] is applied. PR EOS can be

130

expressed as [23],

131

P

RT a  V  b V V  b   b V  b 


(1)

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where P represents pressure; T represents temperature; V represents molar volume; R represents

133

the universal gas constant; and a and b represent the attraction parameter and repulsion parameter,

134

respectively. a and b can be calculated by [23],

135

136

R 2Tc2 a  0.45724  Tr ,   Pc

b  0.07780

(2)

RT Pc

(3)

137

where Pc represents critical pressure; Tc represents critical temperature; ω represents acentric

138

factor; and Tr represents reduced temperature (which can be calculated by Tr 

139

function in Equation (2) can be expressed as [24],

140

141

 1  0.37464  1.54226  0.26992 2 1  T 0.5  2 ,   0.49  r      2  1   0.379642  1.48503  0.164423 2  0.016666 3 1  Tr0.5   ,   0.49  

(4)

For a mixture, the van der Waals mixing rule is used, c

142

T ). The αTc

c

a   xi x j 1  kij  ai a j

(5)

i 1 j 1

c

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b   xi bi

(6)

i 1

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where c represents the number of components in the mixture; x represents component mole fraction;

145

kij represent the binary interaction parameter (BIP) between components i and j; and the subscripts

146

i and j represent components i and j, respectively. The detailed procedure for conducting two-

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phase flash calculation is proposed by Michelsen [25]. 7 ACS Paragon Plus Environment


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For the three-phase VLS equilibrium calculation algorithm proposed by Li and Li [21], the

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asphaltene-precipitation model developed by Nghiem et al. [22] is applied. In this model,

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asphaltene phase is considered as a pure dense phase only containing asphaltene. The fugacity of

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asphaltene in the asphaltene phase is calculated by [22],

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ln f s  ln f  * s

Vs  P  P*  RT

(7)

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where P and P* represent the actual pressure and the reference pressure, respectively; fs and fs*

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represent the fugacities of asphaltene in the asphaltene phase at P and P*, respectively; Vs

155

represents the asphaltene molar volume; and the subscript s represents asphaltene phase. As

156

mentioned above, for vapor and liquid phases, PR EOS [23] is applied.

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Li and Li [21] provided the details of the three-phase VLS equilibrium calculation algorithm. In

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their three-phase VLS equilibrium calculation algorithm, the appearance of asphaltene phase is

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checked by comparing the fugacity of asphaltene component in the non-asphaltene phase against

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that in the asphaltene phase, while the appearance of non-asphaltene phase (i.e., vapor or liquid

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phase) is determined by stability test. The detailed procedure of stability test is proposed by

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Michelsen [26]. If neither of these two cases appears, the tested mixture is stable. If only asplathene

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phase appears, a two-phase vapor/liquid-asphaltene flash calculation is required. If only non-

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asphaltene phase appears, a two-phase vapor-liquid flash calculation needs to be conducted. After

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that, the results yielded by the two-phase flash calculation should be again checked if the

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asphaltene phase will appear. If it appears, a three-phase flash calculation needs to be conducted.

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If both asphaltene and non-asphaltene phases may appear, a three-phase flash calculation is first

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conducted. If the results yielded by the three-phase flash calculation are unphysical, a two-phase

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flash calculation is then conducted. Moreover, the stability of the results yielded by the two-phase 8 ACS Paragon Plus Environment

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flash calculation should be checked. If it is unstable, a three-phase flash calculation should be

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again conducted. Two different three-phase VLS flash calculation algorithms are developed and

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incorporated into the three-phase equilibrium calculation algorithm to ensure its robustness and

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efficiency. To simulate CO2 flooding in light oil reservoirs, new initialization approaches for both

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stability tests and flash calculations are proposed in their work [21].

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3. MMC Algorithm Considering Asphaltene-Precipitation Effect

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Figure 1 depicts the schematic of the contact method adopted by the MMC method that considers

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the asphaltene-precipitation effect. This MMC algorithm is a modification of the MMC algorithm

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developed by Ahmadi and Johns [18]. The detailed procedure for conducting this modified MMC

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algorithm is summarized as follows,

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(1) Fill two cells with the injection gas (G) and reservoir oil (O), respectively. Put the cell filled

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with the injection gas in the upstream, while put the cell filled with the reservoir oil in the

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downstream.

183 184 185

(2) Mix the injection gas and the reservoir oil. The composition of the mixture can be calculated by the following equation,

Z i  Oi   (Gi  Oi ), i  1,..., c

(8)

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where Zi, Gi, and Oi represent the mole fractions of component i in the gas-oil mixture, injection

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gas, and oil, respectively; α represents the mole fraction of gas phase; c represents the number

188

of components; and the subscript i represents the component index. As mentioned by Ahmadi

189

and Johns [18], the value of α will not affect the final results. In our tests, we changed the value

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of α from 0.1 to 0.9 with a step size of 0.1 and found that the value of α will not affect the

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calculated minimum tie line length. In our algorithm, α is set to be 0.5.



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(3) Conduct three-phase VLS equilibrium calculation on the gas-oil mixture at the given

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temperature/pressure condition. The three-phase VLS equilibrium calculation algorithm

194

developed by Li and Li [21] is applied in this MMC algorithm. The asphaltene-precipitation

195

model used in their algorithm is the one developed by Nghiem et al. [22]. In that model,

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Nghiem et al. [22] assumed that the asphaltene phase only contain asphaltene component.

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Based on such assumption, the composition of asphaltene phase can be simply expressed as

198

follows,

1, i  c Si   0, i  c

199

(9)

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where Si represents the mole fraction of component i in the asphaltene phase and component c

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represents the asphaltene component. The possible phases yielded by the three-phase

202

equilibrium calculation will be an equilibrium liquid phase (X), an equilibrium vapor phase

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(Y), and an asphaltene phase (S). If asphaltene phase appears based on the three-phase

204

equilibrium calculations, we assume that it will deposit and not participate in the subsequent

205

calculations. Therefore, the composition of the mixture can be updated by the following

206

equation,   Z / 1   s  , i  1,..., c  1 Zi   i    Z i   s  / 1   s  , i  c

207

(10)

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where βs represents the mole fraction of asphaltene phase calculated by the three-phase VLS

209

equilibrium calculation algorithm. If the asphaltene phase does not appear, the composition of

210

the mixture will not change.

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(4) Conduct two-phase flash calculation on the gas-oil mixture with the updated composition at

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the given temperature/pressure condition. The negative flash [19, 20] is allowed in this MMC

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algorithm. PR EOS is used in two-phase flash calculations [23]. The two-phase flash 10 ACS Paragon Plus Environment

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calculation will give us an equilibrium liquid phase (X) and an equilibrium vapor phase (Y).

215

Based on the fact that gas moves faster than the liquid, the equilibrium vapor phase will be put

216

ahead of the equilibrium liquid phase. The aforementioned steps show the procedure of the

217

first contact. The tie line length between the equilibrium liquid phase and the equilibrium vapor

218

phase requires to be recorded for each contact. It can be calculated by the following equation, TL 

219

c

  X i  Yi 

2

(11)

i 1

220

where TL represents the tie line length and Xi and Yi represent the mole fractions of component

221

i in the equilibrium liquid phase and equilibrium vapor phase, respectively.

222

(5) There are two contacts in the second contact. One is conducted on the mixture of the injection

223

gas and the equilibrium liquid phase, while the other is conducted on the mixture of the fresh

224

oil and the equilibrium vapor phase. Similar to the first contact, the three-phase VLS

225

equilibrium calculation is first conducted on the mixture to check whether the asphaltene phase

226

will appear or not. If the asphaltene phase appears, the composition of the mixture needs to be

227

updated by excluding the precipitated asphaltene. Two-phase flash calculation is subsequently

228

conducted on the updated mixture. Both of these two contacts will give us an equilibrium liquid

229

phase and an equilibrium vapor phase. Together with the injection gas and the fresh oil, there

230

will be six cells. These cells are schematically shown in Figure 1.

231

(6) Additional contacts between the neighbouring cells are required until all c-1 key tie lines are

232

developed. Note that, in the Nth contact, the contacts will be conducted N times. These contacts

233

will be numbered from 1 to N in a left-to-right order as shown in Figure 1. As mentioned by

234

Ahmadi and Johns [18], the key tie line is considered to be fully developed if the same values

235

of tie line lengths are yielded by three neighbouring contacts. The minimum tie lie length

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should be recorded. 11 ACS Paragon Plus Environment


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Figure 1 Schematic of the contact method adopted by the MMC method considering asphalteneprecipitation effect. G: injection gas; O: oil; X: equilibrium liquid phase; Y: equilibrium vapor phase; S: solid asphaltene phase; and 1st contact, 2nd contact, 3rd contact, …, and Nth contact: contact times.

239 240 241 242 243 244

This MMC algorithm is performed to develop the minimum tie line length at the given

245

temperature/pressure conditions. As shown in Figure 1, in this MMC algorithm, the condensing

246

drive occurs at the downstream, while the vaporizing drive occurs at the upstream. In the middle

247

of this development, the miscibility is developed by a combined condensing and vaporizing drive.

248

To obtain the MMP, the MMC algorithm is required to be conducted at different pressures. As

249

pressure increases, the developed minimum tie line length will decrease. In this work, MMP is

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estimated by the power-law extrapolation using the minimum tie line lengths at the last several

251

pressures by [18],


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TLn  aP  b

252

(12)

253

where n represents the exponential parameter; P represents pressure; and a and b are constants

254

representing the slope and the intercept of y-axis, respectively. The values of a, b, and n can be

255

finalized when the square of the correlation coefficient exceeds 0.999. The MMP is the pressure

256

at which the minimum tie line length becomes zero. This MMC algorithm only relies on the

257

performance of three-phase VLS equilibrium calculations and two-phase flash calculations. In

258

comparison with other methods used to estimate MMP which also relies on relative permeability,

259

volume of cells, and volume of injected gas, the revised MMC algorithm provides a simple way

260

for modeling the effect of asphaltene precipitation on MMP.

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4. Results and Discussion

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In this work, three oil and gas samples from Weyburn reservoir in Saskatchewan, Canada, are used

263

to test the performance of the newly developed MMC algorithm that couples the asphaltene-

264

precipitation effect. Both of the MMPs predicted by our algorithm and those predicted by the MMC

265

algorithm proposed by Ahmadi and Johns [18] without considering asphaltene-precipitation effect

266

are compared with the MMPs measured by slim tube experiments conducted by Dong et al. [27].

267

Table 1 lists the compositions of Fluid W1 and three injection gases [4, 27]. Fluid W1 has a gravity

268

of 29ºAPI [4] and a molecular weight of C6+ of 205 g/mol [27]. The asphaltene content of Fluid

269

W1 is measured to be 4.9 wt% [4]. Table 1 Compositions of Fluid W1 and three injection gases [4, 27]

270

Component

Fluid W1

Gas 1 (pure CO2)

Gas 2 (impure CO2)

Gas 3 (impure CO2)

N2

0.96

0

0

5.1

CO2

0.58

100

90.1

89.8



Page 14 of 42

H2S

0.30

0

0

0

C1

4.49

0

9.9

5.1

C2

2.99

0

0

0

C3

4.75

0

0

0

i-C4

0.81

0

0

0

n-C4

1.92

0

0

0

i-C5

1.27

0

0

0

n-C5

2.19

0

0

0

C6+

79.74

0

0

0

271 272

Table 2 depicts the composition and component properties of Fluid W1. The critical properties and

273

acentric factor of the first ten components are cited from literatures [17, 28], while the mole

274

fractions and component properties of pseudo-components are calculated by Li and Li [21]. Table 2 Composition and component properties of Fluid W1 [17, 21, 28]

275

Component

Mole fraction (mol%)

Molecular Critical Critical weight temperature pressure (g/mol) (K) (bar)

Critical volume (m3/kmol)

Acentric factor

N2

0.96

28.01

126.20

34.40

0.0901

0.04

CO2

0.58

44.01

304.70

74.80

0.0940

0.23

H2S

0.30

34.08

373.60

90.60

0.0976

0.10

C1

4.49

16.04

190.60

46.70

0.0993

0.01

C2

2.99

30.07

305.43

49.50

0.1479

0.10

C3

4.75

44.10

369.80

43.00

0.2029

0.15

i-C4

0.81

58.12

408.10

37.00

0.2627

0.18

n-C4

1.92

58.12

419.50

38.00

0.2547

0.20

i-C5

1.27

72.15

460.40

34.30

0.3058

0.23

n-C5

2.19

72.15

465.90

34.00

0.3040

0.24


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C6-9

27.93

101.12

579.68

34.34

0.3897

0.32

C10-17

29.94

179.18

717.68

23.72

0.6536

0.54

C18-27

14.43

300.79

843.53

17.74

0.9369

0.77

C28A+

5.80

513.00

963.33

14.39

1.1924

0.96

C28B+

1.64

513.00

963.33

14.39

1.1924

0.96

276 277

To apply the asphaltene-precipitation model developed by Nghiem et al. [22] in the three-phase

278

VLS equilibrium calculations, Li and Li [21] tuned some key model parameters in the asphaltene-

279

precipitation model by matching the asphaltene precipitation data measured by Srivastava et al.

280

[4] for Fluid W1. Table 3 summarizes the adjustable parameters and their values adjusted by Li

281

and Li [21] for Fluid W1 mixed with pure CO2.

282 283

Table 3 Adjustable parameters and their adjusted values used in three-phase VLS equilibrium calculation algorithm for Fluid W1 mixed with pure CO2 [21] Parameter Adjusted value Critical gas concentration above which asphaltenes start to precipitate (mol%)

46

θ (exponential parameter in Equation (13))

4.7

BIP between CO2 and HCs

0.07

284 285

Table 4 shows the BIPs used by Li and Li [21] for Fluid W1. The BIPs between N2 and other

286

components (except C28B+) and that between H2S and other components (except C28B+) are

287

taken from literature [28]. The BIPs between asphaltene component (C28B+) and other

288

components are calculated by the Chueh and Prausnitz model [29],

289

 2vci1/6 vcj1/6  kij  1   1/3 1/3   vci  vcj 




(13)


Page 16 of 42

290

where vc represents critical molar volume; θ represents the exponential parameter; and the

291

subscripts i and j represent components i and j, respectively. BIPs between asphaltene component

292

(C28B+) and other components are calculated by the same value of θ. Table 4 BIPs used for Fluid W1 [21, 28] N2 CO2 H2S C28B+

293

N2

0

-0.0170

0.1767

0.3447

CO2

-0.0170

0

0.0974

0.3359

H2S

0.1767

0.0974

0

0.3280

C1

0.0311

0.0700

0.080

0.3245

C2

0.0515

0.0700

0.0833

0.2433

C3

0.0852

0.0700

0.0878

0.1828

i-C4

0.1033

0.0700

0.0474

0.1374

n-C4

0.0800

0.0700

0.0600

0.1426

i-C5

0.0922

0.0700

0.0600

0.1129

n-C5

0.1000

0.0700

0.0600

0.1139

C6-9

0.1000

0.0700

0.0600

0.0780

C10-17

0.1000

0.0700

0.0600

0.0233

C18-27

0.1000

0.0700

0.0600

0.0038

C28A+

0.1000

0.0700

0.0600

0

C28B+

0.3447

0.3359

0.3280

0

294 295

Figure 2 shows the comparison between the weight fractions of the precipitated asphaltenes as a

296

function of gas concentration calculated by Li and Li [21] and those obtained by experiments

297

conducted by Srivastava et al. [4] for Fluid W1 mixed with Gas 1 (pure CO2). The tested

298

temperature and pressure are 59ºC and 160 bar, respectively. As seen in Figure 2, the weight

299

fractions of the precipitated asphaltenes calculated by Li and Li [21] have a good agreement with


Page 17 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


300

those measured by experiments, except the point with the highest asphaltene-precipitation amount.

301

The gas concentration at this point is 75 mol%. As mentioned by Li and Li [21], the weight fraction

302

of asphaltene component in the mixture (comprised of 75 mol% Gas 1 and 25 mol% Fluid W1) is

303

smaller than 3.0 wt%. However, the measured weight fraction of the precipitated asphaltene for

304

this mixture is larger than 3.0 wt%. Such discrepancy leads us to ignore this experimental data

305

point in our calculations.

306 307 308 309

Figure 2 Comparison between the weight fractions of the precipitated asphaltenes as a function of gas concentration calculated by Li and Li [21] and those obtained by experiments conducted by Srivastava et al. [4] for Fluid W1 mixed with Gas 1 (pure CO2) at 59ºC and 160 bar

310 311

Figure 3 shows the pressure-composition (P-X) diagram generated by Li and Li [21] for Fluid W1

312

mixed with Gas 1 at 59ºC. As mentioned by Li and Li [21], this P-X diagram is generated by

313

running the three-phase VLS equilibrium calculation algorithm (proposed in their work) a total of 17 ACS Paragon Plus Environment


314

475,209 times at different pressures and gas concentrations. As shown in Figure 1, their algorithm

315

performs well at all the tested conditions. Note that no asphaltene precipitates at lower pressures.

316

For Fluid W1 mixed with Gas 2 and 3, similar P-X diagrams are generated by Li and Li [21].

317 318 319 320

Figure 3 P-X diagram generated by Li and Li [21] for Fluid W1 mixed with Gas 1 (pure CO2) at 59ºC. V: vapor phase; L: liquid phase; S: solid asphaltene phase; and Region 1: three-phase VLS equilibrium region where the liquid phase is very dense and more like an asphaltene phase.

321 322

4.1 Case Study 1: Fluid W1 Displaced by Gas 1 (Pure CO2)

323

Figure 4 shows the variation of the minimum tie lie lengths at different pressures for Fluid W1

324

displaced by Gas 1 at 59ºC. Figure 4(a) depicts the comparison of the minimum tie lie lengths with

325

and without considering the asphaltene-precipitation effect. In Figure 4(a), the black line

326

represents the results calculated by the MMC method developed by Ahmadi and Johns [18]

327

without considering asphaltene-precipitation effect, while the red line presents the results yielded

328

by our MMC algorithm considering asphaltene-precipitation effect. As shown in Figure 4(a), at 18 ACS Paragon Plus Environment

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329

lower pressures (where no asphaltene precipitates), the minimum tie line lengths calculated by our

330

algorithm are the same as those calculated by the MMC method proposed by Ahmadi and Johns

331

[18]; while at higher pressures (where asphaltene precipitates), the minimum tie line lengths

332

calculated by our algorithm are smaller than those calculated by the MMC method proposed by

333

Ahmadi and Johns [18]. Moreover, MMP yielded by our MMC algorithm (which is 145.4 bar) is

334

larger than that yielded by the algorithm proposed by Ahmadi and Johns [18] (which is 132.8 bar).

335

These MMPs are estimated by the power-law extrapolation as shown in Equation (12). Figures

336

4(b) and 4(c) present the extrapolation of the minimum tie lie lengths yielded by the MMC method

337

proposed by Ahmadi and Johns [18] and that yielded by the MMC method proposed by this work,

338

respectively. In Figures 4(b) and 4(c), the blue dots represent the results calculated by the

339

algorithms at the last several pressures. R2 represents the square of the correlation coefficient,

340

indicating how close these calculated points are to the straight line. When R2 equals to 1, these

341

calculated TLs have a perfect linear relationship with pressure. As mentioned previously, a power-

342

law extrapolation (as shown in Equation (12)) is used to estimate the MMPs. We consider the TLn

343

and P have a perfect linear relationship when R2 exceeds 0.999. The finalized exponential

344

parameter (n) used in Figure 4(b) is 4.3, while that used in Figure 4(c) is 3.5.



345 346

(a)

347 348

(b) 20 ACS Paragon Plus Environment

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349 350 351 352 353 354 355 356 357

(c) Figure 4 Variation of the minimum tie lie lengths as a function of pressure for Fluid W1 displaced by Gas 1 at 59ºC: (a) comparison of the minimum tie lie lengths yielded by the MMC method proposed by Ahmadi and Johns [18] and those yielded by the MMC method proposed by this work; (b) extrapolation of the minimum tie lie lengths yielded by the MMC method proposed by Ahmadi and Johns [18]; and (c) extrapolation of the minimum tie lie lengths yielded by the MMC method proposed by this work.

358

Figure 5 depicts the variation of the tie line length calculated by our MMC algorithm as a function

359

of contact number for Fluid W1 displaced by Gas 1 at 59ºC and 110 bar. Four profiles of tie line

360

lengths yielded at different contact times are shown in Figure 5. As previously mentioned, the key

361

tie line can be considered to be fully developed if the same values of tie line lengths are yielded

362

by three neighbouring contacts [18]. This means that the key tie line will be developed in Figure

363

5 when there is a zero slope among three neighbouring contacts. As seen in Figure 5, the key tie

364

lines are gradually developed as contact time increases, demonstrating that our algorithm preforms

365

well in detecting the key tie lines. The shortest key tie line is nearly developed after the 25th contact, 21 ACS Paragon Plus Environment


366

whereas it is considered to be fully developed after the 100th contact. As shown in Figure 5, the

367

shortest key tie line is the crossover tie line in the middle of the contacts, indicating that the

368

miscibility of Fluid W1 and Gas 1 will be generated by a combined condensing and vaporizing

369

drive.

370 371 372 373 374

4.2 Case Study 2: Fluid W1 Displaced by Gas 2 (Impure CO2)

375

Figure 6 shows the variation of the minimum tie lie lengths as a function of pressure for Fluid W1

376


377

and without considering the asphaltene-precipitation effect. Similar to Figure 4(a), the same

378

minimum tie line lengths are calculated by these two algorithms at lower pressures (where no

379

asphaltene precipitates); while at higher pressures (where asphaltenes precipitate), smaller

Figure 5 Variation of the tie line length calculated by our MMC algorithm as a function of contact number for Fluid W1 displaced by Gas 1 at 59ºC and 110 bar


Page 22 of 42

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380

minimum tie line lengths are calculated by our algorithm compared with those calculated by the

381

MMC method proposed by Ahmadi and Johns [18]. Again, our MMC algorithm predicts a larger

382

MMP (i.e., 177.9 bar) than that predicted by the algorithm proposed by Ahmadi and Johns [18]

383

(i.e.,166.9 bar). The power-law extrapolation as shown in Equation (12) is adopted to predict these

384

MMPs. Figures 6(b) and 6(c) present the extrapolation of the minimum tie lie lengths yielded by

385

the MMC method proposed by Ahmadi and Johns [18] and that yielded by the MMC method

386

proposed by this work, respectively. The exponential parameters (n) used in Figures 6(b) and 6(c)

387

are 3.8 and 2.5, respectively.

388 389

(a)



390 391

(b)

392 393

(c) 24 ACS Paragon Plus Environment

Page 24 of 42

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Figure 6 Variation of the minimum tie lie lengths as a function of pressure for Fluid W1 displaced by Gas 2 at 59ºC: (a) comparison of the minimum tie lie lengths yielded by the MMC method proposed by Ahmadi and Johns [18] and those yielded by the MMC method proposed by this work; (b) extrapolation of the minimum tie lie lengths yielded by the MMC method proposed by Ahmadi and Johns [18]; and (c) extrapolation of the minimum tie lie lengths yielded by the MMC method proposed by this work.

394 395 396 397 398 399 400 401

Figure 7 presents the variation of the tie line length calculated by our MMC algorithm as a function

402

of contact number for Fluid W1 displaced by Gas 2 at 59ºC and 110 bar. Figure 7 contains profiles

403

of tie line lengths generated at different contact times. As contact time increases, the key tie lines

404

are gradually detected. As shown in Figure 7, at the 25th contact, the shortest key tie line is almost

405

developed. In this calculation, we consider the shortest key tie line to be fully developed at the

406

100th contact. Same as shown in Figure 5, the shortest key tie line appears at the middle of this

407

development, indicating that a combined condensing and vaporizing drive dominates the gas

408

flooding process. The shortest key tie line length in Figure 7 is larger than the counterpart shown

409

in Figure 5 that is generated under the same temperature/pressure condition for Fluid W1 displaced

410

by Gas 1 (pure CO2). Therefore, a larger pressure will be required in order to obtain the key tie

411

line with a zero length for Fluid W1 displaced by Gas 2 (impure CO2). This implies that the

412

presence of CH4 in CO2 will increase the MMP between CO2 and crude oil.



413 414 415 416 417

Figure 8(a) shows the variation of the weight fractions of precipitated asphaltene calculated by our

418

MMC algorithm as a function of contact number for Fluid W1 displaced by Gas 2 at 59ºC and 110

419

bar. Profiles of the weight fractions of precipitated asphaltene yielded by different contact times

420

are presented in Figure 8(a). Figure 8(b) depicts the weight fractions of precipitated asphaltene and

421

the tie line lengths calculated by our MMC algorithm as a function of contact number when the

422

total contact number is 100. The weight fraction of precipitated asphaltene shown in Figure 8 is

423

calculated by dividing the weight of precipitated asphaltene by the weight of the mixture (obtained

424

after mixing the neighbouring cells). As seen in Figure 8(a), asphaltene only precipitates in the

425

middle of the displacement. Moreover, as shown in Figure 8(b), asphaltene precipitation occurs

426

near the contact number where the shortest tie line is developed. Thus, the asphaltene precipitation

427

will have an obvious effect on the minimum tie line length.



Page 26 of 42

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428 429


(a)

430



(b) Figure 8 Fluid W1 displaced by Gas 2 at 59ºC and 110 bar: (a) Variation of the weight fractions of precipitated asphaltene calculated by our MMC algorithm as a function of contact number in different contact times. (b) Weight fractions of precipitated asphaltene and tie line lengths calculated by our MMC algorithm as a function of contact number when the total contact number is 100.

431 432 433 434 435 436 437 438 439

4.3 Case Study 3: Fluid W1 Displaced by Gas 3 (Impure CO2)

440

Figure 9 shows the variation of the minimum tie lie lengths as a function of pressure for Fluid W1

441


442

and without considering the asphaltene-precipitation effect. Again, at lower pressures (where no

443

asphaltene precipitates), the same minimum tie line lengths are calculated by these two algorithms,

444

while at higher pressures (where asphaltene precipitates), smaller minimum tie line lengths are

445

calculated by our algorithm compared with those calculated by the MMC method proposed by

446

Ahmadi and Johns [18]. Similar to the findings made in case studies 1 and 2, MMP yielded by our

447

MMC algorithm (which is 216.7 bar) is larger than that yielded by the algorithm proposed by

448

Ahmadi and Johns [18] (which is 189.3 bar). Equation (12) is applied to estimate the MMPs.

449

Figures 9(b) and 9(c) shows the extrapolation of the minimum tie lie lengths yielded by the MMC

450

method proposed by Ahmadi and Johns [18] and that yielded by the MMC method proposed by

451

this work, respectively. The exponential parameter (n) used in Figures 9(b) and 9(c) are 3.5 and

452

1.5, respectively.


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453 454

(a)

455 456

(b) 29 ACS Paragon Plus Environment


457 458 459 460 461 462 463 464 465 466

Figure 10 depicts the variation of the tie line length calculated by our MMC algorithm as a function

467

of contact number for Fluid W1 displaced by Gas 3 at 59ºC and 110 bar. Profiles of the tie line

468

lengths developed by different contact times are summarized in Figure 10. Figure 10 shows the

469

gradual development of the key tie lines as contact times increases. The shortest key tie line almost

470

appears after the 25th contact, whereas the fully development of the shortest key tie line is

471

considered to occur after the 100th contact. As shown in Figure 10, the development of the shortest

472

key tie line occurs in the middle of the displacement. Again, this indicates that the miscibility

473

between Fluid W1 and Gas 3 is developed by a combined condensing and vaporizing drive. The

(c) Figure 9 Variation of minimum tie lie lengths as a function of pressure for Fluid W1 displaced by Gas 3 at 59ºC: (a) comparison of the minimum tie lie lengths yielded by the MMC method proposed by Ahmadi and Johns [18] and those yielded by the MMC method proposed by this work; (b) extrapolation of the minimum tie lie lengths yielded by the MMC method proposed by Ahmadi and Johns [18]; and (c) extrapolation of the minimum tie lie lengths yielded by the MMC method proposed by this work.


Page 30 of 42

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474

shortest key tie line length in Figure 10 is larger than the counterpart shown in Figure 5, indicating

475

that the key tie line with a zero length will be yielded at a larger pressure for Fluid W1 displaced

476

by Gas 3 (impure CO2). Therefore, we can conclude that the presence of N2 and CH4 in CO2 will

477

increase the MMP between CO2 and crude oil.

478 479 480 481 482

Figure 11 depicts the variation of the vapor-liquid equilibrium ratios of each component calculated

483

by the two-phase flash calculation in our MMC algorithm as a function of contact number for Fluid

484

W1 displaced by Gas 3 at 59ºC at different pressures. The tested pressures for Figures 11(a), 11(b),

485

11(c), and 11(d) are 90 bar, 110 bar, 170 bar, and 193 bar, respectively. These four figures in

486

Figure 11 are all generated at the 100th contact. The vapor-liquid equilibrium ratios can be

487

calculated by,




488

Ki 

Yi , i  1,..., c Xi

Page 32 of 42

(14)

489

where Ki represents vapor-liquid equilibrium ratio of component i. When the vapor-liquid

490

equilibrium ratios move closer to 1, the compositions of the equilibrium vapor and liquid phases

491

are closer to each other, leading to a smaller tie line length. When the vapor-liquid equilibrium

492

ratios of all components equal to 1, the tie line length will become zero, indicating that the injection

493

gas and the crude oil are miscible. As seen from Figure 11, the values of the equilibrium ratios

494

move closer to 1 as pressure increases. The equilibrium ratios are clustering towards 1 at the

495

contact number where the shortest tie line is developed.

496 497

(a) 32 ACS Paragon Plus Environment

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498 499


(b)



500 501

(c)


Page 34 of 42

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502 503 504 505 506 507 508

4.4 Comparison of Measured and Predicted MMPs

509

Table 5 compares the measured and predicted MMPs for Fluid W1 displaced by three gases at

510

59ºC. The absolute relative error shown in Table 5 is calculated by,

511

(d) Figure 11 Variation of the vapor-liquid equilibrium ratios calculated by the two-phase flash calculation in our MMC algorithm as a function of contact number for Fluid W1 displaced by Gas 3 at 59ºC: (a) 90 bar; (b) 110 bar; (c) 170 bar; and (d) 193 bar.

ARE 

MMPmeasured  MMPpredicted MMPmeasured

100%

(14)

512

where ARE represents absolute relative error (ARE) between the measured and predicted MMPs,

513

MMPmeasured and MMPpredicted represent the measured and predicted MMPs, respectively. The 35 ACS Paragon Plus Environment


514

experimental data are measured by Dong et al. [27] through slim tube experiments. As shown in

515

Table 5, for Fluid W1 displaced by Gas 2 and Gas 3, the MMPs predicted by the algorithm

516

proposed by this work are larger than the measured ones, while the MMPs predicted by the

517

algorithm proposed by Ahmadi and Johns [18] are smaller than the measured ones. For these two

518

cases, the MMPs predicted by our algorithm are closer to the measured ones compared with those

519

predicted by the algorithm proposed by Ahmadi and Johns [18], indicating that our algorithm can

520

provide a more accurate predicted MMP when asphaltene precipitates during the CO2 flooding

521

process. However, for Fluid W1 displaced by Gas 1, both of the predicted MMPs are larger than

522

the measured one. Moreover, for this case, the ARE between the MMP predicted by Ahmadi and

523

Johns [18] and the measured one is smaller than that between the MMP predicted by this work and

524

the measured one. This may be caused by many reasons, e.g., the values of the key parameters

525

used in the three-phase VLS equilibrium calculation algorithm, the characterization of the

526

undefined cut, the BIPs used, EOS used, etc. These factors will not only influence the three-phase

527

VLS equilibria predicted by the VLS algorithm but also the MMPs predicted by the MMC

528

algorithm. As shown in Figure 2, discrepancies still exist between the calculated amounts of the

529

precipitated asphaltenes and the measured ones. By modifying these factors, we may obtain a

530

better description of the asphaltene-precipitation behavior and have a more accurate prediction of

531

the MMPs. Note that, as seen in Table 5, all MMPs predicted by our algorithm considering the

532

asphaltene-precipitation effect are larger than those predicted by the algorithm proposed by

533

Ahmadi and Johns [18] without considering the asphaltene-precipitation effect. This is consistent

534

with the experimental study that asphaltene precipitation tends to increase MMP [5]. As previously

535

mentioned, Kariman Moghaddam and Saeedi Dehaghani [17] proposed an algorithm to predict

536

MMPs with the consideration of asphaltene-precipitation effect by extending the MMC method


Page 36 of 42

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537

developed by Jaubert et al. [15]. Table 5 summarizes the MMPs predicted by Kariman

538

Moghaddam and Saeedi Dehaghani [17] as well. As shown in Table 5, all the MMPs predicted by

539

Kariman Moghaddam and Saeedi Dehaghani [17] are smaller than those predicted by our

540

algorithm. The ARE of the algorithm developed by Kariman Moghaddam and Saeedi Dehaghani

541

[17] is smaller than that of our algorithm for Fluid W1 displaced by Gas 1, while the AREs of the

542

algorithm developed by Kariman Moghaddam and Saeedi Dehaghani [17] are larger than those of

543

our algorithm for Fluid W1 displaced by Gas 2 and 3. Moreover, as can be seen in Table 5, the

544

measured and predicted MMPs for Fluid W1 displaced by impure CO2 (Gas 2 and Gas 3) are larger

545

than those for Fluid W1 displaced by pure CO2 (Gas 1), indicating that the presence of N2 and CH4

546

in CO2 will increase the MMP between CO2 and crude oil.

547 548

Table 5 Comparison of measured and predicted MMPs for Fluid W1 displaced by three gases at 59ºC Measured MMP (bar) [26]

Predicted MMP (bar)

Slim tube

Algorithm proposed by this work

Algorithm proposed by Ahmadi and Johns [18]

128 175 212

145.4 177.9 216.7

132.8 166.9 189.3

Injection gas

Gas 1 Gas 2 Gas 3

Algorithm proposed by Kariman Moghaddam and Saeedi Dehaghani [17] 128.2 159.1 192.8

ARE (%)

Algorithm proposed by this work

Algorithm proposed by Ahmadi and Johns [18]

13.59 1.66 2.22

3.75 4.63 10.71

Algorithm proposed by Kariman Moghaddam and Saeedi Dehaghani [17] 0.16 9.09 9.06

549 550

5. Conclusions

551

In this study, we develop an MMC algorithm considering asphaltene-precipitation effect by

552

modifying the MMC method proposed by Ahmadi and Johns [18]. The main modification is that

553

the three-phase VLS equilibrium calculation algorithm developed by Li and Li [21] is conducted

554

before each contact to check if the asphaltene phase will appear. If asphaltene phase appears, the 37 ACS Paragon Plus Environment


555

composition of the mixture of the fluids in two neighbouring cells needs to be updated by excluding

556

the asphaltene phase. The two-phase flash calculation should be conducted on the mixture with

557

the updated composition. Both the MMPs predicted by our algorithm and those predicted by the

558

MMC algorithm without considering asphaltene-precipitation effect are compared with the MMPs

559

measured by slim tube experiments conducted by Dong et al. [27] for three real reservoir fluids.

560

The comparison results show that the MMPs predicted by our algorithm are larger than those

561

predicted by the MMC algorithm without considering asphaltene-precipitation effect for all tested

562

examples, which is the same as the experimental results that the asphaltene precipitation will

563

increase MMPs. Moreover, for Fluid W1 displaced by Gas 2 and Gas 3, the MMPs predicted by

564

our algorithm are closer to the MMPs measured by experiments compared with the MMPs

565

predicted by the MMC algorithm without considering asphaltene-precipitation effect. For Fluid

566

W1 displaced by Gas 1, the MMP predicted by the original MMC algorithm without considering

567

the asphaltene-precipitation effect is closer to the measured one compared to the MMP predicted

568

by our new MMC algorithm; and both of the predicted MMPs by the original MMC algorithm and

569

our new MMC algorithm are larger than the measured one.

570

Acknowledgements

571

The authors greatly acknowledge a Discovery Grant from the Natural Sciences and Engineering

572

Research Council (NSERC) of Canada to H. Li.

573

Nomenclature a

attraction parameter in PR EOS or slope

b

repulsion parameter in PR EOS or intercept of y-axis

c

number of components or asphaltene component

error

absolute relative error between measured and predicted MMPS 38 ACS Paragon Plus Environment

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f

fugacity

Gi

mole fraction of component i in the injection gas

k

BIP

Ki

vapor-liquid equilibrium ratio of component i

MMP

MMP

n

exponential parameter

Oi

component mole fraction in oil

R

universal gas constant

S

component mole fraction in asphaltene phase

TL

tie line length

v

molar volume

V

molar volume

x

component mole fraction in mixture

X

component mole fraction in equilibrium liquid phase

Y

component mole fraction in equilibrium vapor phase

Z

component mole fraction in gas-oil mixture

Greek Letters α

gas phase mole fraction or α-function

β

phase mole fraction

ω

acentric factor

θ

exponential parameter

Superscript *

reference parameter

Subscript c

critical property 39 ACS Paragon Plus Environment


i

component index

j

component index

measured

measured value

predicted

predicted value

r

reduced parameter

s

asphaltene phase

574 575

References

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A Modified Multiple-Mixing-Cell Algorithm for Minimum Miscibility

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